= Degree of a continuous mapping
{title2=$\deg f$}
{wiki}
= Mapping degree
{synonym}
For $m\geq1$, a <continuous map> $f:S^m\to S^m$ induces multiplication by one <integer> on $H_m(S^m;\mathbb Z)\cong\mathbb Z$. That integer is its mapping degree:
$$
f_*[S^m]=(\deg f)[S^m].
$$
It records the signed number of times the domain wraps around the target, made precise by the <degree as a sum of local degrees>. The degree is invariant under <homotopy>, and <multiplicativity of mapping degree> follows by composing the induced <homomorphisms>.
The <circle> map $z\mapsto z^r$ has degree $r$ for every <integer> $r$. The <antipodal map> on $S^m$ has degree $(-1)^{m+1}$. A constant map has degree zero. The same fundamental-class definition gives the <degree of a map between oriented manifolds> for arbitrary <connected> <closed manifolds> of equal <dimension> with chosen <orientations>. For $S^0$, use <reduced homology> instead of its rank-two ordinary $H_0$; the <degrees of maps of the zero-sphere> are only $-1,0,1$.
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